Adjoint divisors on algebraic curves

Let C be a numerically connected curve lying on a smooth algebraic surface. We show that if 4 is an ample invertible sheaf satisfying some technical numerical hypotheses then omega(C) circle times A is normally generated. As a corollary we show that the sheaf omega(C)(circle times2) on a numerically connected curve C of arithmetic genus p(a) greater than or equal to 3 is normally generated if omega(C) is ample and does not exist a subcurve B subset of C such that p(a) (B) = I = B(C - B). (C) 2003 Elsevier Inc. All rights reserved.